One of the essential approximations of allab initiomethods is the introduction of abasis set. Expanding the molecular orbitals in a set of known functions is not an approximation if the basis set iscomplete. A complete basis set means using aninfinitenumber of functions, which is impossible. Then afinitebasis set must
2.4 - Basis Sets 53 be used, but taking into account that the smaller basis set will provide a poorer electronic representation. The type of basis functions used also influences the accuracy.
Modern electronic structure methods can be classified into two classes, depend-ing on the choice of the basis set: localized basis functions with Gaussian-type orbitals (GTOs) and plane wave methods.
2.4.1 Gaussian-Type Orbitals
For non-periodic systems, such as molecules, Gaussian type functions are the natural choice, because they give a reasonable description of the physics of the system and their integration is computationally easy. The generic molecular orbital can thus be expressed asφi=PM
α cα,iχα, whereχαis anatomic orbital. There are two types of atom-centered basis functions commonly used in electronic structure calculations: Slater Type Orbitals (STO)199and Gaussian Type Orbitals (GTO).200 Slater type orbitals have the functional form as:
χ(r, θ, ϕ) =N Yl,m(θ, ϕ)rn−1e−ζr (2.23) However, resolution of the bielectronic integrals is easier using Gaussian functions (GTOs):
χ(r, θ, ϕ) =N Yl,m(θ, ϕ)r2n−2−le−ζr2 (2.24) Where, N is a numerical factor tonormalizethe function to unity andl, m, nare integers that characterize the type and order of the Gaussian function.
2.4.2 Plane waves
Plane waves201represent another common choice of basis functions, particularly in solid-state simulations. They are periodic functions that can be written in
54 Chapter 2 - Theoretical Background terms of complex exponentials or sine-cosine functions. For instance, for the free electron in one dimension:
φ(x) =Aeikx+Be−ikx φ(x) =Acos(kx) +Bsin(kx)
(2.25)
For infinite systems, the molecular orbitals coalesce intobands. The electrons in a band can be described by orbitals expanded in a basis set ofplane waves, which in three dimensions can be written as a complex function:
χk(~r) =eik·r (2.26) Regarding Bloch’s theorem198with plane waves basis functions the wavefunction of an electron within a perfectly periodic potential may be written as:
ψi,k(r) =ui(r)eik·r (2.27) whereui(r)is a function that adds the periodicity of the potential asui(r+l) = ui(r), wherelis the length of the unit cell;kis a wavevector confined to the first Brillouin Zone. Sinceuj(r)is a periodic function, we may expand it in terms of a Fourier series:
ui=X
G
ci,GeiG·r (2.28)
whereGare thereciprocal latticevectors defined throughG·R= 2πm, where mis an integer;Ris a real space lattice vector and theci,Gare the plane wave expansion coefficients. The electron wavefunctions may therefore be written as a linear combination of plane waves so-calledBloch functions.
ψi,k(r) =X
G
ci,k+Gei(k+G)·r (2.29) In principle, the series in the previous equation should be infinite, but in practice the series should be truncated in order that it may be handled computationally.
The coefficient for the plane waves have a kinetic energy 2m~2 |k+G|2, and plane waves with high kinetic energy are less important than those of low kinetic energy.
2.4 - Basis Sets 55 Then, a kineticenergy cut-off Ecuthas to be introduced in order to achieve a finite basis set. The kinetic energy cut-off is defined through:
Ecut= ~2
2m |k+G|2 (2.30)
fixing the highest reciprocal lattice vectorGused in the plane wave expansion, resulting in a finite basis set. A typical energy cutoff of 200 eV on a cubic unit cell ofa= 15Åcorresponds to a basis set with20.0·103functions. Plane wave basis sets tend to be significantly larger than the typical Gaussian-type basis sets.
Furthermore, basis-set superposition errors that have to be carefully controlled in calculations based on local basis sets are avoided with plane wave functions.
They are the best choice for describing delocalized slowly varying electron densi-ties, such as the valence and conduction bands in metals. The core electrons are strongly localized around the nuclei, which means that to describe them adequately a large number of plane wave basis is needed. The nuclei-electron potential is furthermore impossible to be described in a plane wave basis, and this type of basis set is often used in connection withpseudopotentialsfor smearing the nuclear charge and model the effect of the core electrons.
2.4.3 Pseudopotentials
Chemical systems involving atoms from the lower part of the periodic table present a large number of core electrons. Moreover, the discussion above points to the fact that large energy cutoffs must be used. The most important approach to reduce the computational cost due to core electrons is to usepseudopotentials.
Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density that matches various important physical and mathematical properties of the true ion core. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential.202
56 Chapter 2 - Theoretical Background
Projector Augmented Wave (PAW) Method
Theprojector-augmented wave(PAW)203was introduced by Blöchl to achieve simultaneously the efficiency ofpseudopotentialsmethod and the accuracy of the full-potential linearized augmented-plane-wave (FLAPW) method, commonly used as benchmark for DFT calculations on solids. The (PAW) method accounts for the nodal features of the valence orbitals and ensures the orthogonality between valence and core wave functions instead of the pure pseudo-potential approach.
The (PAW) wave function is represented as a valence term expanded in a plane-wave basis adding the contribution from the region within the core radius of each nucleus, evaluated on a grid. The contribution from a core region is expanded as a difference between two sets of densities, one arising from the all-electron atomic orbitals, the other from a set of nodeless pseudo-atomic orbitals, allowing the wave function within the core region to adjust for different environments.